Instituto de Física Gleb Wataghin
Débora Princepe
Optical gain medium incorporation into
semiconductor optomechanical cavities
Incorporação de meio de ganho óptico em cavidades
optomecânicas de semicondutor
Campinas
2018
Optical gain medium incorporation into
semiconductor optomechanical cavities
Incorporação de meio de ganho óptico em cavidades
optomecânicas de semicondutor
Thesis presented to the “Gleb Wataghin” Institute of Physics of the University of Campinas in partial fulfilment of the requirements for the degree of Doctor in Science.
Tese apresentada ao Programa de Pós Graduação em Física do Instituto de Física “Gleb Wataghin” da Universidade Estadual de Campinas para obtenção do título de Doutora em Ciências.
Supervisor/Orientador: Newton Cesário Frateschi
Co-supervisor/Co-orientador: Gustavo Silva Wiederhecker
Este exemplar corresponde à versão final da tese defendida pela aluna Débora Princepe e orientada pelo Prof. Dr. Newton Cesário Frateschi
Campinas
2018
ORCID: 0000-0002-1158-1993
Ficha catalográfica
Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174
Princepe, Débora,
P935o PriOptical gain medium incorporation into semiconductor optomechanical cavities / Débora Princepe. – Campinas, SP : [s.n.], 2018.
PriOrientador: Newton Cesário Frateschi. PriCoorientador: Gustavo Silva Wiederhecker.
PriTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.
Pri1. Laser de semicondutor. 2. Optomecânica de cavidade. 3. Semicondutores III-V. 4. Nanofotônica. I. Frateschi, Newton Cesário, 1962-. II. Wiederhecker, Gustavo Silva, 1981-. III. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Incorporação de meio de ganho óptico em cavidades
optomecânicas de semicondutor Palavras-chave em inglês: Semiconductor lasers Cavity optomechanics III-V Semiconductors Nanophotonics
Área de concentração: Física Titulação: Doutora em Ciências Banca examinadora:
Newton Cesário Frateschi [Orientador] Felippe Alexandre Silva Barbosa Lucas Heitzmann Gabrielli Paulo Alberto Nussenzveig Ben-Hur Viana Borges
Data de defesa: 30-01-2018
Programa de Pós-Graduação: Física
MEMBROS DA COMISSÃO JULGADORA DA TESE DE DOUTORADO DE DÉBORA PRINCEPE– RA: 70601 APRESENTADA E APROVADA AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 30/01/2018.
COMISSÃO JULGADORA:
- Prof. Dr. Newton Cesario Frateschi- (Orientador) - IFGW/UNICAMP - Prof. Dr. Felippe Alexandre Silva Barbosa- IFGW/UNICAMP - Prof. Dr. Lucas Heitzmann Gabrielli- FEEC/UNICAMP
- Prof. Dr. Paulo Alberto Nussenzveig – INSTITUTO DE FÍSICA - USP
- Prof. Dr. Ben-hur Viana Borges– ESCOLA DE ENGENHARIA DE SÃO CARLOS
A Ata de Defesa, assinada pelos membros da Comissão Examinadora, consta no processo de vida acadêmica do aluno.
CAMPINAS 2018
Ao terminar essa etapa, quero agradecer a todas as pessoas que estiveram presentes em minha vida. Tive muita ajuda desde os primeiros passos e hoje, na realização da minha tese, sinto alegria em agradecê-las.
Em primeiro lugar, à minha família, a quem dedico esse trabalho e tenho amor incondicional. Mãe, pai, irmã e irmão, cada um deles, à sua maneira, me ajudou de forma fundamental para chegar até aqui. Ainda que somos tão diferentes, amo vocês.
A todos os professores que tive na vida, pela inspiração e conhecimento transmitido. Em especial, ao Newton, pela orientação de quase uma década e por ensinar o valor do foco e do trabalho duro, sempre com alegria. Também ao Gustavo, que dividiu a orientação dessa tese, e em quem vi o esmero e a dedicação que espero ter na minha carreira. Ainda agradeço a outros professores que me ajudaram nesse processo: Thiago, dando ótimas sugestões, e Ivan, que me recebeu tão bem em seu laboratório, desenvolvendo ciência de qualidade com incrível entusiasmo – Merci beaucoup pour tout! Aos colegas de pós-graduação, amigos (dear friends from France too) e ex-parceiros, pela companhia, descontração e aprendizado. Tive oportunidade de dividir aulas, salas e laboratórios com pessoas extremamente talentosas, com quem refleti temas além da ciência. Às pessoas com quem dividi casa, agradeço pelo enfretamento da minha incoveniente individualidade. Felizmente guardo muitas dessas amizades, o bastante para não listar e correr risco de ser injusta. Muito obrigada pela constante desconstrução e por me ajudarem a ser uma pessoa melhor.
A todos os funcionários da Universidade, em especial do Instituto de Física e do Departamento de Física Aplicada. O trabalho da equipe técnica do LPD e do CCS Nano foi essencial, assim como das secretarias de Graduação, Pós-Graduação e Financeira. À OSA e ao capítulo de estudantes da OSA na Unicamp, pela oportunidade de desenvolver atividades extra-curriculares tão recompensadoras.
A todas as mulheres antes de mim, que lutaram para que eu tivesse o direito de estudar, de entrar em uma Universidade e de defender um doutorado.
À Universidade Estadual de Campinas, que possibilitou minha formação como licenciada, bacharel e agora doutora em física.
Enfim agradeço às agências Fapesp (2011/18945-6 e 2014/25453-0) e CNPq (143359/2011-8) que financiaram esse trabalho.
but I think I have ended up where I needed to be.
Sistemas optomecânicos e eletromecânicos baseados em semicondutores III-V tornaram-se um tópico de interesse. Além das propriedades atrativas desses materiais, essa atenção deve-se principalmente à possibilidade de integração do oscilador optomecânico e da cavidade do laser em um único dispositivo. A investigação teórica e experimental mostra que a interação nesses sistemas híbridos pode levar ao aumento da taxa de resfriamento optomecânico, controle da emissão laser, entre outros efeitos.
Nesta tese, apresentamos nosso trabalho em osciladores optomecânicos ativos de material semicondutor. Sob uma abordagem semi-clássica, foi desenvolvido um modelo baseado em equações de taxa de laser acopladas a um oscilador harmônico, em que tanto as ressonâncias como as perdas ópticas são modificadas pela deformação da cavidade. Esse modelo difere da optomecânica de cavidades usual dado que o laser de bombeio e a dessintonia estão ausentes, levando a um controle feito pela injeção de corrente. O acoplamento entre vibração mecânica e oscilação de relaxação do laser é derivado da dinâmica do sistema, de modo que o regime de amplificação é alcançado para determinado valor de acoplamento optomecânico global. Sob essa condição, os fótons e a vibração mecânica apresentam oscilação auto-sustentada – obtém-se então um laser auto-pulsado. A instrumentação para fabricação e medida do laser optomecânico foi desenvolvida com base em dispositivo microdisco com bombeio óptico com acoplamento optomecânico puramente dispersivo. Apresentamos então a investigação dos parâmetros relevantes para o projeto da cavidade em plataformas de GaAs. A emissão laser e a interação optomecânica são previstas e observadas em microdiscos com meio de ganho de poço quântico. Discutimos os desafios envolvidos na obtenção de um laser optomecânico. Finalmente, apresentamos o conceito de um laser optomecânico realista com otimização do acoplamento optomecânico empregando os mecanismos dispersivo e dissipativo. Essencialmente, um bullseye optomecânico ativo com forte acoplamento dispersivo é combinado com uma estrutura dissipativa, um anel metálico externo. A interação dispersiva é intensificada pelo confinamento de modos mecânicos e ópticos perto da borda do disco e o efeito de acoplamento dissipativo resulta da interação de campo próximo com o anel metálico separado por um estreito espaçamento. Mostra-se que esse novo dispositivo optomecânico dissipativo potencialmente resulta em um laser optomecânico realista.
Optomechanical and electromechanical systems based on III-V semiconductor materials have become a topic of high interest. Beyond the attractive material properties, this attention is mainly due to the possibility of integrating an optomechanical resonator with a laser cavity in a single device. Theoretical and experimental investigation show that interaction in these hybrid systems can provide enhancement of the optomechanical cooling rate, control of the laser emission, among other effects.
In this thesis, we present our work on active optomechanical resonators built on semiconductor material. Under a semi-classical approach, a model based on the laser rate equations coupled to a harmonic oscillator was developed, with both optical resonance and loss modified by the cavity deformation. This model differs from usual Cavity Optomechanics since both the driving field and the detuning are absent, leading to a control done by current injection. A novel coupling model between mechanical vibration and laser relaxation oscillation is derived from the system dynamics, such that the amplification regime is achieved for certain value of overall optomechanical strength. Under this condition, photons and mechanical vibration present self-sustained oscillation – therefore, a self-pulsed laser is obtained.
Instrumentation for the fabrication and measurement of an optomechanical laser was developed based on microdisk geometry with optical pump and purely dispersive optomechanical coupling. We then present an investigation of the relevant parameters for the design of an optomechanical laser cavity based on GaAs platform. Laser emission and optomechanical interaction are predicted and observed in microdisks with quantum well gain medium. We discuss the significant challenges involved in obtaining an optomechanical laser.
Finally, we present the design of a realistic optomechanical laser with a strong enhancement of the optomechanical coupling employing both dispersive and dissipative mechanisms. Essentially an active optomechanical bullseye with very strong dispersive coupling is combined with a dissipative structure, an external metallic ring. The dispersive interaction is enhanced by the confinement of mechanical and optical modes near the disk edge and the dissipative coupling effect is provided by near field interaction with the metallic ring separated by a small gap. This novel dissipative optomechanical device is shown to potentially result in a realistic optomechanical laser.
BCK Mixture of Hydrobromic Acid, Acetic Acid and Potassium Dichromate
BS Beam Splitter
CB / VB Conduction Band / Valence Band
CW Continuous Wave
DBR Distributed Bragg Reflector DFB Distributed Feedback DH Double Heterostructure ENBW Effective Noise Bandwidth ESA Electrical Spectrum Analyzer FEM Finite Element Method FSR Full Spectral Range
FWHM Full Width at Half Maximum HCG High Contrast Grating
HSQ Hydrogen Silsesquioxane IR / NIR Infrared / Near Infrared LP / SP Long-Pass / Short-Pass
MB Moving Boundary
MBE Molecular-Beam Epitaxy
MEMS/NEMS Microelectromechanical / Nanoelectromechanical System MOCVD Metallo-Organic Chemical Vapor Deposition
MQW Multiple Quantum Wells OSA Optical Spectrum Analyzer PE Photoelastic Effect
PL Photoluminescence
QD Quantum Dot
QW Quantum Well
RBW Resolution Bandwidth
ROF Relaxation Oscillation Frequency
RF Radio-Frequency
RT Room Temperature
SEM Scanning Electron Microscopy SQW Single Quantum Well
SVEA Slow Varying Envelope Approximation TE / TM Transverse Electric / Transverse Magnetic VCSEL Vertical-Cavity Surface-Emitting Laser WGM Whispering Gallery Mode
0 Vacuum permittivity µ0 Vacuum permeability
c Vacuum speed of light: c =qµ1
00
~ Normalized Planck’s constant kB Boltzmann’s constant
q Elementary charge m0 Free electron mass i Imaginary unity ˆ
x, ˆy, ˆz Unit vectors in Cartesian space Ωcav Cavity resonance frequency
κi Intrinsic optical decay rate
κe External decay rate due to radiation loss and input coupling κ Overall cavity intensity decay rate: κ = κi+ κe
Qo Optical quality factor: Qo = Ωcav/κ x Mechanical motion amplitude Ωm Mechanical frequency
Γm Mechanical damping
Qm Mechanical quality factor: Qm= Ωm/Γm meff Effective motional mass
xzpf Mechanical zero-point fluctuation amplitude: xzpf =q~/(2meffΩm) ΩL Input laser frequency
gω Dispersive optomechanical coupling factor: gω = ∂Ωcav/∂x
gκ Dissipative optomechanical coupling factor: gκ = ∂κ/∂x
G Overall hybrid optomechanical coupling strength: G2 = gωgκx2zpfP
a Complex light amplitude, normalized to photon number: a= hˆai, a∗a=Dˆa†ˆaE= P
δΩm Optical spring (mechanical frequency shift) Γom Optomechanical damping
V Optical cavity volume
p Intracavity photon density, total number: P = pV ¯n Charge carrier density
α Absorption coefficient β Spontaneous emission factor σ Medium conductivity
χ Susceptibility
Material’s complex dielectric constant ζ Linewidth enhancement factor
k0 Vacuum wavenumber
˜k Material’s propagation constant: ˜k = k0√ ˜nref Complex refractive index
n0 Cold cavity refractive index
vg Group velocity of light in a medium: vg = c/(n0(ω) + ω∂n0 ∂ω)
p Photon density ¯n Carrier density G Optical gain
¯
G Net gain rate: ¯G= ΓvgG ¯
G¯n Net differential gain: vg∂G∂¯n ¯
G0 Net gain rate at steady state Γ Confinement factor
Bsp Bimolecular recombination coefficient Rsp Spontaneous emission rate: Rsp= Bsp¯n2 Rst Stimulated emission rate: Rst = ¯Gp I (Ith) Injected current (threshold current)
¯ntr Carrier density for transparency
¯nth Carrier density in the threshold
ΓN Carrier fluctuation damping
Resumo 7
Abstract 8
1 Why active optomechanical cavities? 16
1.1 Active optomechanical resonators – present status . . . 18
1.2 Our contribution . . . 22
2 Fundamental background 25 2.1 Semiconductor laser . . . 25
2.1.1 Design of a laser cavity . . . 27
2.1.2 Laser rate equations . . . 29
2.1.3 Steady state and dynamic analysis . . . 35
2.2 Cavity optomechanics . . . 38
2.2.1 Optical and mechanical motion equations . . . 39
2.2.2 Optomechanical coupling and dynamical backaction . . . 40
2.2.3 Dissipative optomechanical systems . . . 44
2.3 Overview remarks . . . 45
3 Model for active optomechanical resonator 47 3.1 How to couple laser light and motion? . . . 47
3.2 Generalized laser rate equations . . . 50
3.3 Small signal analysis . . . 52
3.4 Laser and mechanical parameters trade-off . . . 55
4 Active microdisk design 57 4.1 Microdisks as starting devices . . . 57
4.2 Optical modes . . . 58
4.3 Mechanical modes . . . 61
4.4 Optomechanical coupling rate . . . 63
5.1.1 Electronic lithography . . . 69
5.1.2 Disk and mesa etching . . . 71
5.1.3 Release and surface treatment . . . 72
5.1.4 Issues of the fabrication process . . . 73
5.2 Devices characterization . . . 74
5.2.1 Optomechanical experiments at transparency . . . 74
5.2.2 Photoluminescence with spatial resolution . . . 78
5.3 Experimental challenges and next steps . . . 83
6 Bullseye optomechanical laser 86 6.1 Active bullseye optomechanical resonator . . . 86
6.1.1 Optical modes and SQW gain medium . . . 88
6.1.2 Mechanical modes . . . 89
6.2 Optomechanical coupling rates . . . 90
6.2.1 Calculation of the dissipative coupling . . . 91
6.3 Results of the model . . . 93
7 Conclusion and Perspectives 95
List of publications 98
Bibliography 99
A Gain of a quantum well 109
B Propagation constant and refractive index in a semiconductor 119 C Optical force with dispersive and dissipative couplings 121
Chapter 1
Why active optomechanical cavities?
The development of hybrid micro and nano-systems coupled to mechanical resonators has become an important topic of theoretical and experimental investigation. Such schemes are based on the transduction of the mechanical motion to the coupled physical system, which may be an optical cavity, an electric or a superconducting circuit, among others. The different forms of coupling mechanisms and geometries resulted in a large variety of configurations to exploit this interaction and led to the development of a wide range of applications [1–8]. In particular, the merging of a mechanical resonator to an optical cavity in the so called field of Cavity Optomechanics gave rise to a successful source of versatile devices for basic and applied research in optics [9, 10]. Among the multiple applications, we can cite tunable filters [11], radio frequency oscillators [12], wavelength converters [13] and optical delay lines [14], besides fundamental applications, such as cooling of the mechanical resonator [15,16].The most explored mechanism to connect the optical and mechanical resonators in these applications is called dispersive coupling and it consists on the optical frequency shift due to the mechanical deformation. The simplest sketch to illustrate this interaction consists of a Fabry-Perot cavity with a movable end mirror, as illustrated in Fig. 1.1(a). In this representation, one of the mirrors is free to dislocate, described by a harmonic oscillator with natural frequency Ωm and intrinsic damping Γm. The optical
cavity may present several resonances and they depend on the cavity size; for a given optical mode, the resonance frequency is Ωcav and the total decay rate is κ. In the
presence of an input laser, photons populate this cavity mode and exert optical force on the mirror walls – the known effect of radiation pressure [17] –, driving the mechanical oscillator. Since the resonance frequency depends on the length of the cavity, the mirror motion induces fluctuation in Ωcav; in the other hand, this causes variation in the optical
force, generating a feedback loop.
A major reason for the investigation of these devices is the effect called dynamical backaction [18]. It consists on the modification of the mechanical susceptibility, i.e. variation on Ωm and Γm, promoted by the delay between the optical force and the
Ωm,Γm Ωcav, κ
laser input
optical
cavity mechanical resonator (a) (b) Ωm Ωm (i) Mo de densit y (a.u.) Frequency Ωm Ωcav ΩL Mechanical Mode Cavity Mode Ωm Ωm (ii) Mo de densit y (a.u.) Frequency Ωm ΩL Ωcav
Figure 1.1: Dispersive cavity optomechanics scheme. (a) Fabry-Perot cavity with resonance Ωcav and decay rate κ. The end mirror is free to oscillate in its normal mode with frequency Ωm and damping Γm. The intracavity photon population is fed by an input laser and exerts optical force on the mirror, driving the mechanical motion. Change in the end mirror position alters the frequency Ωcav, generating a feedback loop. (b) Scattering picture of heating (i) and cooling (ii) of the mechanical state. In (i), the laser ΩL is blue detuned from the cavity resonance, and then the Stokes sideband scattering is favored. The exceeding energy is added to the mechanical mode, which is possible when the detuning is the same amount or multiple of Ωm. The opposite situation in (ii), when the laser is red detuned, favors the anti-Stokes scattering and it is necessary to take energy out from the mechanical mode, cooling it.
oscillator response due to the finite nature of the cavity lifetime. The force component in phase with the motion promotes change in Ωm, modifying the resonator stiffness,
denominated optical spring effect. The out-of-phase component induces then variation of Γm, called optomechanical damping rate. We say that the mechanical mode is heated
(or amplified) when Γm is decreased, storing more energy, and it is cooled when Γm
increases, having energy taken out of the mode – in the quantum mechanical picture of the harmonic oscillator, these correspond respectively to creation and annihilation of phonons of the vibrational mode.
The control of this delay is done through the laser input which pumps the optical cavity, with light at frequency ΩL. When this beam is mismatched from the cavity
resonance, energy is exchanged with the mechanical resonator to adjust the photons to the cavity frequency. The scattering picture, represented in Fig. 1.1(b), summarizes how the
exchange between optical and mechanical energy occurs. Basically, the laser response has sidebands due to the mechanical modulation of the mirror, ΩL−Ωm (Stokes) and ΩL+Ωm
(anti-Stokes). When the laser is blue detuned from Ωcav, as in (i), the band with lower
frequency is favored. If this detuning is equal to or a multiple of the mechanical frequency, the optomechanical coupling allows the mechanical mode to receive the exceeding energy, increasing the phonon number, i.e. the motion is amplified. In (ii), when the laser is in the opposite side of the resonance, the anti-Stokes scattering is then favored, and it is necessary to take energy from the mechanical oscillator to populate the cavity with higher energy photons. Then phonons are destroyed and the mechanical state is said to be cooled. Such mechanism can be applied to any other geometry of optomechanical cavity, as long as it is possible to establish a coupling between the optical frequency and the mechanical degree of freedom.
The early development of optomechanical resonators was essentially carried out in passive devices: these systems are pumped by a coherent input channel, a driving laser as described above, which is built and controlled externally, while the cavity itself presents very low absorption and no emission – equivalent to a high optical quality factor, defined as Qo = Ωcav/κ. The idea to integrate the light source emerged naturally, which
means to build a laser cavity with a mechanical degree of freedom. One important aspect of such device is the additional light-matter interaction that generates the laser light and may produce novel optomechanical phenomena as well as relax the conditions for some known phenomena observed in passive cavities. In this context, the pursuit of active optomechanical cavities is a relevant topic for research on optomechanics in cavities.
1.1
Active
optomechanical
resonators
–
present
status
Theoretical work has been done in optomechanical cavities incorporating material with optical gain. Given the prominent demonstration of cooling of the mechanical state, the first efforts were focused on this kind of application, in order to optimize the cooling rate through the optical transition. This remarkable interest relies mainly on the goal for realization of the ground state, with relevance for quantum physics related research [3,19]. A precursor scheme with this purpose was suggested in [1], where it was shown that the coupling of a mechanical resonator to a quantum dot (QD) presents assisted cooling provided by optical absorption, likely to achieve the zero phonon state. The coupling between the dot and mechanical mode occurs directly through the strain in the material, which modifies the band structure and hence the QD levels. Following this idea, assisted cooling could be facilitated by the confinement of a cavity, promoting the optomechanical interaction and the optical feedback necessary to later build a laser.
Physical modeling of an active optomechanical cavity have to consider hypothesis about the gain parameters, which depend on the nature of the embedded medium – diverse gain profiles are obtained from the available optical materials, generating distinct effects. A very simplified way to introduce absorption and gain is to consider an atomic ensemble, modeled as a two-level system with narrow bandwidth transition. This general scheme is usual in the treatment of these hybrid systems, and a recurrent motivation in these works is to eliminate the known condition of resolved sideband operation [20, 21]. This regime happens when the mechanical frequency exceeds the linewidth of the cavity mode, Ωm κ, such that the scattering unbalance is very sharp,
as the ideal representation in Fig. 1.1(b). The optical resonator is then called good cavity, optimal to perform efficient cooling [22]. For real systems, however, this is not always satisfied. When the system is not sideband resolved, both amplification and cooling are less efficient, since the sidebands have almost the same density of states inside the mode profile and then the enhancement of one of them and suppression of the other is not as effective.
A pioneer work in this aspect was done by Genes, Ritch, and Vitali, in order to relax this requirement [20]. Given the scattering representation for the good cavity (Fig. 1.2(a)), it is possible to obtain the same scheme in bad cavities by modifying them with atomic ensemble, with absorption in ω1 = ωs(Fig. 1.2(b)) or with inverted population
with transition ω2 = ωas (Fig. 1.2(c)). The limitations for this demonstration are how
narrow the transition has to be and the need of fine control of the two-level population, of whose value the cooling rate depends. Other posterior works brought similar treatment to achieve ground state cooling via auxiliary media [23, 24], however, they do not configure yet laser cavities, since there is no optical feedback to provide stimulated emission.
Figure 1.2: Introduction of narrow bandwidth gain or loss provided by atomic ensemble to relax the resolved sideband condition for cooling [20]. (a) In a good cavity, with resonance ωc, the mode profile (dashed black line) scatters the driving laser ωl to the sidebands ωs and ωas favoring the anti-Stokes sideband. For a bad cavity, there are two possibilities to reproduce this scheme: (b) Transition at ω1 = ωs allows absorption and suppresses the lower sideband. (c) Inverted population with ω2= ωas adds gain, promoting enhancement of the upper sideband.
Thereafter a more general question was raised about the possibility of a laser cavity to perform cooling on its own mirrors, eliminating the need of an external driving laser. This problem is approached in [25] and it is shown that this is not possible: considering a cavity with atomic transition and usual dispersive coupling, the optomechanical damping rate tends to zero over the laser threshold. In fact, when it is fed by incoherent pump, the atomic medium itself acts as an amplifier, generating extra noise and thus always heating the mechanical resonator. Nevertheless, when the external driving signal is reintroduced at the emission frequency, the pump intensity tunes the optomechanical rate and it can be larger than the one obtained in a passive cavity, implying in more efficient cooling.
The ongoing idealization and modeling of these systems represent the relevance of this subject. However, active optomechanical cavities performing cooling were not yet experimentally demonstrated. Meanwhile, one shall notice the possibility to explore the opposite effect, the amplification of the mechanical motion in active cavities. Also, other aspects of the system may produce interesting effects, such as the intrinsic nonlinearity, which gives rise to bistable behavior [24], the balance between gain and loss in coupled cavities [26–28], or other forms of gain, like the ones provided by bulk semiconductor material or quantum wells (QWs) [29]. In fact, when it comes to the recent experimental investigation of active optomechanical cavities, different possibilities were explored rather than cooling. For experimental purposes, it was very convenient to use the already known geometries and structures explored for laser cavities in the past, choosing the ones that allowed to add a mechanical degree of freedom. In this perspective, the demonstrated effects were in most of the time result of each system particularities, but still with very interesting applications.
Among the recently explored designs, we find the relevant examples of cavities based on the “zipper” geometry and planar devices formed by distributed Bragg reflectors (DBRs). In the first case, the photonic crystal double-clamped nanobeam, called zipper cavity, was reproduced on a semiconductor laser structure and integrated to electric contacts, which capacitively act on the double beam, as shown in Fig. 1.3(a) [30]. A tunable optically pumped laser was demonstrated, where the electromechanical actuation resonantly excites mechanical modes, inducing the optical frequency shift. Related to DBR planar microcavities, also known as VCSELs (acronym for vertical-cavity surface-emitting laser), Fig. 1.3(b) and (c) show very different applications. In [31], Czerniuk et al. implemented a planar microcavity exhibiting simultaneous confinement of both optical and mechanical resonances due to the Bragg reflectors, which work as mirrors and also create stop bands in the phonon dispersion. When the gain medium is incorporated, QWs or QDs, the laser light couples to the phonon oscillation and their strong interaction induces modulation of the output intensity. Such demonstration was performed with optical pump and the mechanical vibration was excited by a strain pulse,
as represented in Fig. 1.3(b). Another explored possibility with a VCSEL is to attach the mechanical oscillator on the top of the device with an empty gap, extending the optical cavity length to the oscillator’s position (Fig. 1.3(c)). In fact, similar construction of microelectromechanical (MEM) mirrors is already largely explored to build tunable lasers with external cavity, even at commercial scale [32]. It was shown that the use of a high contrast grating, or HCG, as end mirror allows faster tuning speed [33], at the same time it changes the emission properties, such as the laser output linewidth [34]. The optimization of the HCG as a mechanical resonator allowed to achieve self sustained oscillation due to the combined effects of radiation pressure and phototermal forces. Using a tuning voltage, enhanced wavelength sweep and large oscillation amplitude were also demonstrated [35].
(a) (b)
(c)
Figure 1.3: Demonstrations of laser microcavities with electromechanical and optomechanical actuation. (a) Wavelength tunable zipper cavity, optically pumped and with mechanical resonant excitation by electrostatic actuators [30]. (b) Spatially coincident resonances of photons and phonons in a DBR planar cavity. The mechanical waves are driven by strain pulses and couples to the laser emission, promoting modulation of it [31]. (c) VCSEL coupled to a high contrast grating (HCG). The membrane vibrational mode is electrostatically excited, changing the emission wavelength and achieving self sustained oscillation [35].
The practical application aspect of the experimental research on laser optomechanics is very significant, as well it shows the fruitful research still being developed on semiconductor lasers. However, we emphasize that the obtained results highly depend on each device’s particularities. There was not yet a general model that could be applied to any geometry and thus be able to predict similar effects in different types of cavity, since the supported mechanical mode and its excitation method are individual to each kind of demonstrated design. Nevertheless, it is clear that cavities based on semiconductor are very convenient for the realization of active optomechanical resonators, due to the versatility of design and well developed experimental infra-structure. III-V compounds stand out as suitable materials: there is a large variety of alloys, allowing emission in a wide region of the spectrum, and the gain can be obtained in different profiles by
incorporation of bulk material, quantum wells or a quantum dots array. Also, it is possible to reproduce successful geometries developed on silicon platforms due to the similarity of mechanical properties and microfabrication techniques. Even beyond the actual laser operation, it is possible to use the absorption and the carrier dynamics in different applications: to change spectroscopy properties of the device [36], to manage the interaction between photons and free carriers [37–39] or to explore the strong coupling regime, commonly achieved in the light-matter interaction in these systems [40,41], among others. Besides that, the mechanics can be incorporated through different mechanisms, such as piezoelectricity [36], photoelastic effect [42] and moving boundary coupling [43].
1.2
Our contribution
In this context, we present the work of this thesis on the development of an active optomechanical resonator, dubbed optomechanical laser, based on III-V semiconductor platform. Our main goal was to understand and demonstrate the interaction between the laser cavity emission and mechanical motion in order to predict novel phenomena and applications of such a system. Given demonstrations on optomechanical systems in the radio frequency range [44–46], we were specially interested in obtaining light emission modulated by optomechanical oscillation, connecting photonic circuits and RF applications. Also, our group, the Device Research Laboratory (Laboratório de Pesquisa em Dispositivos – LPD), has conducted the research in both semiconductor lasers [47–51] and optomechanics in cavities [52–55], such that this doctoral work took advantage of the pre-existing knowledge and contributed for the development of a brand new device.
Based on the usual laser rate equations, we propose a semi-classical model to couple the laser oscillation and the mechanical motion. In this approach, such interaction is enabled when the cavity presents both the optical resonance and decay rate modulated by the mechanical displacement – called respectively dispersive and dissipative optomechanical couplings. There was not similar proposal regarding active optomechanical devices so far, standing out for the novelty character of this approach. As result, we have shown that self sustained oscillation of the motion is achieved when the cavity fulfills some requirements concerning mechanical and optoelectronic parameters. The emission intensity presents modulation in this regime, provided by the coupling between the vibrational mode and the laser relaxation oscillation. The control is done through the pump intensity or injected current, building up for optomechanics without the external drive laser and the usual figure of merit of the detuning.
At the light of this analysis, we aimed to develop an optomechanical laser cavity for demonstration of this effect. We started from the microdisk resonator geometry, explored for both laser cavities [56–58] and optomechanical oscillators [59–61]. Such
optical cavity sustains typical whispering gallery modes (WGM), allowing evanescent coupling with an integrated waveguide or a suspended fiber [62]. The disk also sustains mechanical modes and the range of frequencies is established by the disk radius. The initial device was optically pumped, due to the simpler epitaxial structure design and experimental realization, allowing the modification for electrical contact [48, 56]. From such simple geometry we could develop fabrication techniques and characterization setups, improving the infra-structure for later more complex structures.
Our model was developed simultaneously to the experimental work and demanded modifications in the cavity design, in order to enable the dissipative coupling and the enhanced optomechanical coupling strength, which we found required to observe the self pulsed emission. Inspired by the recent demonstration of the bullseye optomechanical resonator in our group [53], we suggest the incorporation of gain medium in this device, specifically quantum wells, to build an optomechanical laser. Its geometry consists on a microdisk with a nanostructured top surface, forming a circular grating optimal for engineering of the mechanical modes dispersion. The design of band gaps for longitudinal waves allows confinement of the vibration to the disk edge, spatially coincident to the optical mode localization, thus promoting enhanced dispersive optomechanical coupling. Our modification to promote the dissipative coupling consisted in the addition of a highly absorptive structure in the vicinity of the disk, typically a metallic ring, forming a gap with the disk sidewall. While the optical shift occurs mainly due to the photoelastic effect, the modulation of the decay rate is essentially governed by the moving boundary contribution: the disk edge displacement perturbatively modifies the air spacing from the metal, changing the near field pattern of the optical mode and its quality factor. Together with the model, this design is a main contribution of this thesis, exploring different mechanisms of optomechanical coupling and mechanical excitation to allow the predicted effect.
Given the motivation for this research, the thesis is organized as follows. In Chapter 2, we do an overview of our work basis: semiconductor lasers and optomechanics in cavities. Both topics are already largely studied in the literature, thus we briefly make a review over the main aspects that are going to be relevant to build our model, right after in Chapter 3. There we present our semi-classical approach to treat the optomechanical laser considering a III-V semiconductor platform and cavity with dispersive and dissipative optomechanical couplings, discussing the highlights and limitations of this model, as well the results on self pulsed emission.
Experimental work was realized in parallel to the theoretical investigation, based on the instrumentation development for fabrication and measurement of compatible cavities. For that, we built active optomechanical microdisks based on gallium arsenide platform, in collaboration with the group of Dr. Ivan Favero from the MPQ Laboratory (Laboratoire Matériaux et Phénomènes Quantiques) in the Université Paris Diderot. In
Chapter 4 we present the design of these devices, by performing simulations of the optical and mechanical modes and calculating the expected optomechanical coupling and optoelectronic properties. The following chapter 5 is then dedicated to the experimental activities. We have obtained GaAs active disks with optical gain provided by embedded single quantum well. We show the fabrication and characterization of these disks, following techniques well established by Dr. Favero’s team, with typical optomechanical measurements and evaluation of the laser performance.
In Chapter 6, we present the active bullseye, our theoretical proposal of optimized device for demonstration of the predicted effects, yet experimentally feasible. As we did in Chapter 4, we discuss the cavity’s design details, in particular the engineering for enhancement of the dissipative optomechanical coupling.
Finally, in Chapter 7 we present our final considerations and perspectives of this work.
Chapter 2
Fundamental background
Our work is based on two well established topics in optics of microcavities: semiconductor lasers and cavity optomechanics. In this chapter, we present a brief review of each subject, focusing on the main aspects necessary to build our model, presented in the next chapter.
2.1
Semiconductor laser
The operating principle of a laser is based on the three fundamental mechanisms of light-matter interaction: absorption, spontaneous and stimulated emission. The understanding about these phenomena was fundamental to the realization of microwaves’ amplification in the 1950’s, followed by the referred demonstration with light. Such amplification in a laser1 is promoted by two essential ingredients: gain and optical
feedback. The gain medium is responsible for converting the energy delivered to the system, called pump, into stimulated emission, while feedback is provided by a cavity, confining light in an optical mode – photons then cross the material multiple times, leading to the amplification and producing a monochromatic and coherent output beam. There is a wide range of laser types classified by the gain medium nature: atomic gases, doped crystals, free electron sources, to cite a few. Besides these, semiconductor materials stand out for their versatility – diode lasers based on these materials represent the most commercialized type, mainly due to the low production cost and efficiency provided by the direct current injection for pumping. With low energy consumption and reduced size, they have become essential for advances in telecommunication, among other industrial applications.
Semiconductor lasers were first introduced by using p-n junctions to obtain stimulated emission. The first demonstrations were based on GaAs devices [63–65], with optical gain provided by electron-hole recombination in the depletion region and cavity
formed by the polished facets in the junction’s endings. Right after, improvement of both carrier and light confinement was suggested by joining materials with different bandgaps, called heterostructure (in contrast with the single material diodes, named homojunctions). Based on this, the double heterostructure (DH) was composed by two materials grown into a sandwich, where the material with wider bandgap forms the outer layers (the claddings) and the second one, with smaller bandgap, is in the middle, as seen in Fig. 2.1. This kind of structure provided the first lasers working at room temperature, enabled by higher efficiency in carrier confinement, resulting in lower threshold current [66].
1 2 Epitaxial growth Band Energy Refractiv e Index Bandgap _ _ _ _ _ + + + + + ∆n=n2-n1 1 VB CB Mode profile
Figure 2.1: Scheme of a double heterostructure: the cladding material (1) has wider bandgap than the middle layer (2), promoting confinement of electrons in the conduction band (CB) and holes in the valence band (VB). Also, the middle layer has higher refractive index, n2 > n1, inducing localization of the optical mode due to the index contrast. (2) is called active region, defined as the region where carriers recombine, producing gain and emitting photons.
The great success of these devices was mostly due to the following progress in telecommunication and fiber transmission systems, where the small size, low consumption and high durability were very important for enlargement of the network. Special development occurred on III-V semiconductor lasers with emission at 1.3 µm and 1.5 µm, corresponding respectively to minimum loss and dispersion in optical fibers. Their functionality go beyond as monochromatic and coherent light sources and have become essential devices for optical signal processing. Surely there is still room for research, in both basic physics and practical applications. The use of coupled cavities to evaluate the trade off between gain and absorption in the laser behavior is an example of recent fundamental investigation, based on the parity-time symmetry exhibited in these systems [67, 68]. Concerning more technological applications, it stands out the demonstration of nanolasers [51], development of integrated platforms with Si for hybrid Si-III-V lasers [69], tunable sources and fast response lasers for modulation [33], and so on.
2.1.1
Design of a laser cavity
A laser cavity must provide simultaneous optical gain and feedback, as mentioned before. Regarding the gain medium, the importance of DH for the development of semiconductor lasers was brought up since it enabled practical applications. A step forward was implemented on reducing the gain layer thickness, in the order of hundreds of Angstroms, such that carriers are subject to quantum effects of the confinement. In this way, the conduction and the valence bands configure quantum wells for electrons and holes respectively, discretizing the energy levels. As a consequence, the carrier density of states is modified and devices with smaller threshold current were obtained [66]. The emitted photon’s energy in this case is equal to the difference between the levels, which depend on the quantum well thickness and its depth, i.e. the band discontinuity. Thus the emission wavelength can be set by the well properties, which is an advance compared to the bulk material for the laser design. This is also tailorable by handling the composition, which intrinsically changes the bandgaps of the well and barrier. Growth of very thin layers with high quality was possible back then due to epitaxial growth techniques developed in the 1970’s, with two main methods: molecular-beam epitaxy (MBE) [70, 71] and metallo-organic chemical vapor deposition (MOCVD) [72, 73]. Both allowed fine control of the composition, thickness and planarity, specially relevant for realization of III-V DH and QW lasers [66].
Confinement can be further extended: while the quantum well has reduced thickness with the other dimensions kept at bulk scale, quantum wires [74] or dots [75] have two or three dimensions diminished, respectively. Ongoing miniaturization of the active region affects the injection efficiency due to the reduced transversal section (resulting in higher electrical resistance), although still leading to small working currents. Also, due to the modified density of states, these devices allow strong light-matter interaction, of special interest for cavity quantum electrodynamics [76] – it is remarkable that a QD presents delta-like density of states, reproducing an atomic behavior.
Calculation of the optical gain is relevant in the sense it allows to calculate the threshold current of the laser, an important design parameter. For derivation of the optical rates, it is necessary to know the band structure of the junction. Alloys properties – bandgap, lattice constant, effective mass, and other pertinent constants – are easily calculated by interpolation of binary compounds [77]. For the DH, the bands alignment was largely studied [78], including the case of accumulated strain due to lattice mismatching [79], i.e. when the substrate and grown material have different lattice constants. From the band structure, we solve the electrons and holes states, considering the confinement configuration: bulk, QW, QD, etc – see in Fig. 2.2 the representation of typical band structure of a QW. Finally, optical transition rates for absorption, spontaneous and stimulated emission are calculated by Fermi’s Golden Rule,
using the time-dependent perturbation theory. The relation between spontaneous and stimulated emission and calculation of the material gain from the electronic properties are shown in detail in Appendix A. It is demonstrated that for the transition between the excited state E2 to the initial state E1, the gain G(E21, f1, f2) and spontaneous emission
rate Rsp are related by
G(E21, f1, f2) = Rsp
η(E21)vg(1 − e
(E21−∆µ)/kBT) (2.1)
where: fi is the Fermi-Dirac occupation probability of the state i, η(E21) is the photonic
density of states with energy E21 = E2− E1, vg is the group velocity of light in the cavity
and ∆µ is the chemical potential, given by the difference between the quasi-Fermi levels; kB stands for the Boltzmann constant and T for the temperature. Also in Appendix A we discuss the details for each step of the calculation, which also includes evaluation of η(E), the transition matrix element, quasi-Fermi levels and inclusion of the intraband relaxation. Energy L ∆EC ∆EV Eg,barrier Eg,well Etransition ec1 hv1
Figure 2.2: Band structure of quantum well with length L. Barrier and well have energy bandgaps Eg,barrierand Eg,well. The correct alignment in the interface is important to establish the bands discontinuities ∆Ec and ∆Ev, which are the well depths. ec1 and hv1 represent the first discretized levels for electrons in the CB and holes (light or heavy) in the VB respectively; the transition is then given by the difference ec1 − hv1.
Besides the optoelectronic properties, the optical cavity geometry is an important design feature and it must be compatible with the gain medium structure while providing optical confinement. One issue is to produce high reflectivity mirrors that allow the collection of light outside the cavity – this also concerns the divergence and directionality of the output beam. With the decided geometry, it is necessary then to perform the fabrication process on the grown wafer to produce devices of micrometric or even nanometric dimensions. Part of the research consists in improving the laser fabrication, optimizing the steps (lithography, etching, metal and anti-reflective coating deposition etc). Fig. 2.3 shows some examples of cavity geometries and respective
(a)
(b) (c)
Figure 2.3: Examples of laser cavity geometries and respective applications. (a) Edge-emitting ridge laser with high CW output power and improved collimation [82]. (b) Microdisk laser for microwave generation connected to output waveguide [83]. (c) Photonic crystal bandedge membrane laser on silicon platform – proof of concept for hybrid integration [84].
applications. It is observed that mirrors can be formed by several forms: cleaved facets (e.g. ridge), periodic structures for gratings (DBR, including VCSEL, DFB and photonic crystal) or by geometrical construction (disk). Also concerning the design, one must consider the pump method, which can be made through current injection or by optical absorption. In the first case, the laser has ohmic contacts that inject carriers directly in the junction, with recombination occurring in the active region [80]. It is necessary to optimize the cladding doping to enable contact between metal and semiconductor and to get efficient injection and diminished heating – an essential optoelectronic design problem. The second method consists on shining light on the structure, satisfied that the pump photons have energy higher than the bandgap of the active material. Due to absorption, electron-hole pairs are created, similarly to the situation of electric contact. Heating may occur here due to scattering to phonons, since these pairs may be created with higher energy than the transition and, after recombination, this excess can be absorbed by the crystal, exciting the vibrations. In this case, the pump is optimized by choosing wavelength close to the transition; also intermediary cladding layers can be added, to avoid recombination in the surface and to improve confinement of carriers to the active region [81]. This method is more usual at research level, while state-of-the-art devices desirably have electric contacts.
2.1.2
Laser rate equations
The laser behavior, concerning the interaction between the optical field and material gain, can be semi-classically understood through the laser rate equations. Such
formulation is the simplest form of semi-classical theory, in which the electromagnetic field is presented classically by the wave equation while the gain medium is quantized, represented by a two-level system [85] – in this approach, the corpuscular nature of light is disregarded, giving its classical character. This formulation is widely used to explain laser properties in general, allowing one to infer about the steady state, spectral and dynamical properties of the emission. The set of equations in this formulation is usually threefold, describing the time derivatives of the optical field amplitude, medium polarization and population inversion (defined as the difference between the excited and ground state populations). Some hypothesis are assumed to get the final equations, including the slow varying envelope approximation (both field and polarization vary slowly in time compared to the optical frequency), small losses (cavity decay rate and transition linewidth are much smaller than the related optical frequencies) and the resonance approximation (lasing frequency is somewhere between the optical resonance and the transition frequency, which must be close). Considering also linear susceptibility, the polarization is written proportional to the field and we end up with two equations, for inversion population and for photon number, the latter derived from the optical field.
When it comes to semiconductor laser, this derivation is more directly, since the linear susceptibility is assumed in the first steps [86]. The analysis starts from Maxwell’s equations in the absence of free charges:
∇ ×E = −µ0∂H ∂t (2.2a) ∇ ×H = J + ∂D ∂t (2.2b) ∇ ·D = 0 (2.2c) ∇ ·B = 0 (2.2d)
E and H are the electric and magnetic field vectors and the medium response is given by D and B, the electric displacement and magnetic flux density. J is the current density vector, which represents the source for the electromagnetic field. There are the auxiliary definitions (constitutive relations): J = σE , with σ equal to the medium conductivity, D = 0E + P, where P is the electric induced polarization, and B = µ0H . With some
mathematical manipulation2 and taken that ∇·P is negligible, we get the wave equation:
∇2E − σ 0c2 ∂E ∂t − 1 c2 ∂2E ∂t2 = 1 0c2 ∂2P ∂t2 (2.3)
Here we apply the condition for linear susceptibility, which relates polarization and electric field at first order. The temporal dependence is also covered by this condition, assuming that the material has instantaneous response – this holds true for semiconductor material
given that intra-band processes are fast compared to carrier and photon lifetimes (order of tenths of picosecond). We can write then P = 0χE , getting
∇2E − 1
c2
∂2
∂t2(E ) = 0 (2.4)
where we used the definition of the material dielectric constant , related to χ and to the loss represented by σ – for more details, see Appendix B. The solution of the wave equation follows writing the field E in a general form, for a certain chosen polarization ˆe:
E (r, t) = ˆeX
j
Φj(r)Ej(t)e−iωjt (2.5)
where Φj(r) describes the j-mode’s spatial profile, satisfying ∇2Φj(r) + ¯k2jΦj(r) = 0,
with ¯kj =
n0Ωcav,j
c – these are the modes of the “cold cavity”, i.e. the optical solution in
space when there is no pump and the refractive index n0 is purely real. Notice that the
amplitude varies over time, Ej(t), due to the cavity losses, thus the mode is no longer
stationary. Choosing a mode j, such that the sum is only over one term, substitution of 2.5 in equation 2.4 gives Φ(r) " c2 ω 2 + iω2 ∂ ∂t ! − ¯k2 # E(t) + Φ(r)2iω c2 " + i ω ∂ ∂t # dE(t) dt = 0 (2.6) where we omitted the index j and considered the second derivatives negligible, since variations of both E(t) and are much slower than ω (slow varying envelope approximation3). Regarding the dielectric constant, one should take = (r, t), since
it depends on the intrinsic material properties and on the pump. A simplification can be performed taking the spatial average:
hi = Z
Φ∗(r)Φ(r)(r)d3
r (2.7)
where Φ(r) is normalized: R Φ∗(r)Φ(r)d3
r = 1. A limitation of this simplification is the neglected spatial dependence of optical modes and medium, since the field is not uniform in the whole cavity. We also removed the temporal dependence of , whose major consequence is the cavity’s dispersion – in time domain, is affected mainly by the carrier injection and generation of light, both time dependent; nevertheless, this static analysis is still reliable for most of the cases. The equation for E(t) is then
hi c2 ω 2− ¯k2 ! E(t) + 2iω c2 hi dE(t) dt = 0 (2.8)
3SVEA can be defined for a field A(t) as ∂2A ∂t2 ω∂A∂t
; the second time derivative of the field is then: ∂t∂22(E(t)e−iωt) = e−iωt[HH
H ∂2E(t)
∂t2 − 2iω
∂E(t)
The average hi can be replaced by an effective dielectric constant. In Appendix B, it is defined the relation between dielectric constant and complex refractive index: √ = ˜nref, with ˜nref = nr+ ini = (n0+ δn) + i(α/2k0), where n0 is the refractive index in absence of
pump, δn is its carrier-induced variation, α is the absorption coefficient and k0 = ω/c is
the propagation constant in vacuum. Hence hi= (nr+ ini)2 = n20+ 2n0δn+ n0
iαc
ω + O(δn
2) + O(α2) + O(δn α) (2.9)
where δn and α are usually smaller than n0, allowing to ignore higher orders terms. Later,
better approximation is done considering the mode index instead of n0. Also, α must be
taken as mode-absorption, equal to the material absorption summed up with other losses (scattering and mirrors imperfections); taking the material absorption as opposite of gain, we use ¯α = αint+ αmirror− G. Using 2.9 in 2.8, disregarding terms αdEdt and δn
dE dt, we get ω2−Ω2cav+ 2ω2δn n0 + i ω¯αc n0 ! E(t) + 2iωdE(t) dt = 0 (2.10)
Here we see the difference between ω and Ωcav: the first is the field oscillation frequency, i.e.
the laser-mode frequency, initially undetermined, while the second is the cavity resonance, solved in the absence of pump. Since ω nearly coincides with Ωcav, then we can write:
ω2−Ω2cav ∼= 2ω(ω − Ωcav) and finally the rate equation for E(t) becomes:
dE(t) dt = i ω −Ωcav+ ω δn n0 ! E(t) − c n0 ¯α 2E(t) (2.11)
It is useful to separate the amplitude and phase of E(t), such as E(t) = A(t)e−iφ(t),
getting the respective rate equations dA dt = vg G − αtot 2 A(t) (2.12) dφ dt = − ω −Ωcav+ ω δn n0 ! (2.13) where we approximated the phase velocity c/n0 by the group velocity vg (disregarding
the medium dispersion [86]) and wrote the net absorption as gain minus total losses (the adopted convention is net absorption negative for lossy cavity and positive for material with gain). It is seen that the amplitude in 2.12 has growth or decay depending on the balance between gain and losses, which is quite intuitive and mark the threshold condition, where G = αtot. The phase equation shows the balance between the cold
cavity resonance Ωcav and lasing frequency, affected by the carrier-induced index change
δn. This non-exact coincidence generates the effect of frequency chirping: the lasing frequency is usually smaller than the cavity resonance and has to be taken into account
in the analysis of emission spectral linewidth.
The amplitude equation can be converted to a description of the photon number P , given that P = 2
2~ω
R
|E |2d3r and then P ∝ A2. It is usual to use the density p, with P = pV and V equal to the cavity volume. The convenience comes from the connection with the optical intensity |E|2, which is the measurable physical quantity; however, it does not imply the discretized behavior of light, corresponding to an average number. This change of variables allows to get the photon density rate equation:
dp
dt = ( ¯G − κ)p + βRsp (2.14) where we defined the net gain rate ¯G = vgG. If the gain region and optical mode have only partial spatial superposition, the fraction of overlap Γ has to be considered, called the confinement factor: Γ =
R g|Φ|
2d3r R
|Φ|2d3r, with “g” corresponding to the active region. The
photon decay rate is also defined: κ = vg(αint + αmirror), and it can be used to get the
photon lifetime: τP = 1/κ. β is the spontaneous emission factor and corresponds to
the amount of photons generated by spontaneous emission, with rate Rsp, that couple to
the lasing mode. This rate is added phenomenologically, while rigorous treatment of the noise due to this recombination requires the field quantization. Within this semiclassical approach, besides the contribution to photon number, the added noise is taken care of in the laser linewidth derivation.
The connection between gain and linewidth is recovered, defining the so called linewidth enhancement factor: ζ = −2k0∂n∂g/∂Nref/∂N, where N is carrier number. Writing
δn= −(ζ/2k0)δG, with vgδG= ¯G − κ, the phase rate equation is written as dφ
dt = −(ω − Ωcav) + 1
2ζ( ¯G − κ) (2.15)
This means that variation in the gain is always followed by index change, that shifts the mode frequency. As said, this property is very relevant in the analysis of the laser linewidth, since small changes in the gain, due to fluctuations in the carrier number, induces changes of the phase, i.e. contributes to phase noise.
Once obtained the equation for photon density, it is necessary to describe the carrier in time, dependent on the pump – written as a current I in the rate equations, provided by electrical injection or through optical absorption. Analogous to the photon rate equation, it is usual to work with carrier densities ¯n, with total number N = ¯nV . Without loss of generality, we consider that the active region material is intrinsic, ideally undoped, such that electrons and holes densities are equal (if not, the charge balance that guarantees the neutrality has to be accounted for). In a phenomenological framework, we may easily describe d¯n
dt as the difference between injected and recombined carrier rates.
equation, which rules the electronic transport: ∂¯n
∂t = D(∇2¯n) + J/(q d) − R(¯n), where D
is the diffusion coefficient, J is the current density (I divided by the transversal section area), q is the electron charge and d is the active layer thickness. R(¯n) is the total recombination rate, considering radiative and non-radiative processes. Originally, ¯n is function of both space and time and the diffusion depends on the device geometry – for more complicated structures, the diffusion term has to be taken into account. Assuming approximately uniform carrier density in the active region and high injection efficiency4,
the diffusion term is negligible and we get
d¯n dt = J q d− R(¯n) (2.16) In general, R(¯n) is described as R(¯n) = Anr¯n + Bsp¯n2 + CAuger¯n3+ Rst (2.17) The first term refers to the non-radiative recombination rate due to defects, composed of contaminating atoms and dislocations, mostly grown in during the epitaxy. They create a continuum of states in their vicinities, such that an electron or hole may diffuse and get trapped in one of these created levels unbound to the crystal5. This term also accounts
for recombination in surface states, which are similar perturbation caused by the interface between different materials, intrinsically disturbing the band structure. The third term accounts for non-radiative recombination in a different form, known as Auger, where the electron-hole recombination provides energy to excite another carrier, from VB or CB, which relaxes back to equilibrium by transferring the extra energy to phonons and heating. The radiative mechanisms have two forms: spontaneous recombination, represented by the quadratic term Bsp¯n2 = Rsp (Bsp is called bimolecular recombination coefficient),
where electron and hole recombine emitting a photon; and stimulated recombination, with rate Rst = ¯Gp, originated from interaction with an optical field, such that the photons
emitted are coherent. Both phenomena are connected, as explicit in the calculation of gain (Appendix A).
The set formed by 2.14 and 2.16 is know as laser rate equations and describes with accuracy static and dynamic properties of semiconductor laser. Exception is made for ultra fast phenomena in semiconductor material, which have to be treated considering quantum effects and many-body calculations, and so the density-matrix formalism is more appropriate and complete, requiring more elaborate calculation [87].
4The pump term must be multiplied by a coefficient for internal efficiency, concerning the efficacy of conversion of carrier to spontaneous emission.
5In special, there are the so called mid gap states, energy levels formed within the semiconductor bandgap. They act as attractors, realizing trap-assisted recombination, where the electron from VB gets to the CB passing through a mid gap state.
2.1.3
Steady state and dynamic analysis
Important laser parameters, such as the threshold current Ith, come from
handling the set of equations formed by 2.14 and 2.16. To the following analysis, we will disregard non-radiative recombination rates, consider Γ ≈ 1 and use a simplified model for gain, dependent only on carrier number in a linear approximation, ¯G(¯n) = vgG¯n(¯n − ¯ntr), where ¯ntr is the carrier density for transparency (where ¯G(¯ntr) = 0). Then 2.16 becomes
d¯n dt = I q V − Bsp¯n 2− ¯G ¯n(¯n − ¯ntr)p (2.18)
with vgG¯n = ¯G¯n. Steady state properties are obtained by setting the time derivatives to
zero, dp
dt = 0 and d¯n
dt = 0. Below threshold, I < Ith, ¯n is low such that Rst ≈ 0 and then
¯n ∼q I
Bspq V. For photons, considering a single lasing mode, from 2.14 we get
p= βRsp
(κ − ¯G) (2.19)
pis inversely proportional to losses minus gain, deriving the threshold condition ¯Gth= κ, point from where the photon density starts to grow and this relation diverges. Hence, we get ¯nth = ¯ntr+ κ/ ¯G¯n and Ith = Bsp¯n2thqV. In fact, the gain after threshold assumes
fixed value ¯Gth, following the constraint on carrier density ¯n(I > Ith) = ¯nth, due to the balance between injection, spontaneous and stimulated emission: higher injection leads to more recombination events and then the extra carriers are always consumed, clamping the carrier density. Nevertheless, the excess energy due to higher I is transfered to photon number: with gain slightly above the losses, 2.19 shows that the equilibrium for photon density is close to a divergence point and small variations of carrier and gain leads to fast change of p, getting to new steady state value p0. This reflects in the dynamic behavior,
as we will see later on. The steady state problem has to be solved numerically to find the solutions ¯n0 and p0: I q V − Bsp¯n 2 0 − ¯G0p0 = 0 (2.20) βBsp¯n20+ ( ¯G0− κ)p0 = 0 (2.21) Fig. 2.4 shows this typical set of solutions for a microlaser6, with the clamp of carrier
density after threshold followed by rapid increase of photon density with the current.
6The parameters used to obtain the figures 2.4, 2.5 and 2.6 were: disk with 3 µm radius and thickness of 230 nm, with optical resonance at λ = 1550 nm and neff = 3. Laser properties – considering linear approximation for the gain: β = 10−4, Bsp = 10−16m−3/s, αi = 500 m−1 (Qi = 2.5 × 104), Qe = 104, ¯
ntr=7 × 1023m−3, ¯G¯n= 1.5 × 10−20m2(later multiplied by the confinement factor, taken that the optical mode stay in the edge, in a circular crown of 1 µm width); in the dynamic simulation, the ratio I/Ithwas 2.0, such that ΩR/2π = 1313 MHz.
